[PPT] - SET-PARTITION TABLEAUX Tom Halverson Macalester College FPSAC 2019 PowerPoint Presentation

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SET-PARTITION TABLEAUX

Tom Halverson Macalester College FPSAC 2019 Ljubljana July 2, 2019

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Set-Partition Tableaux

Integer Partition:

8, 13 14 2, 3 1, 6, 10 16 4 12 5, 7, 9 15, 17 11

Set Partition: 8 > > < > > : {2, 3}, {4}, {5, 7, 9}, {1, 6, 10}, {11}, {12}, {8, 13}, {14}, {16}, {15, 17} 9 > > = > > ; Organization:

I. Origins: Representation

Theory of the Symmetric Group

II. Schur-Weyl Duality: The

Partition Algebra and Other Diagram Algebras

III. Insertion Bijections

May 2016: [Benkart-H-Harmon] Dimensions of irreducible modules ... [Orellana-Zabrocki] Symmetric group characters as symmetric functions

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I. Symmetric Group Tensor Power Representations

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Origins: The Symmetric Group Sn

I Mn = n-dimensional permutation module ⇠ = Sn Sn

I Basis: v1, v2, . . . , vn with group action: (vi) = vσ(i)

I Sλ

n = irreducible CSn-module, ` n

I M⌦k

n

= k-fold tensor product module

I Diagonal action on basis of simple tensors: (vi1 ⌦ vi2 ⌦ · · · ⌦ vik) = vσ(i1) ⌦ vσ(i2) ⌦ · · · ⌦ vσ(ik)

Question Determine the multiplicity mλ

k,n in the decomposition:

M⌦k

n

= M

λ`n

mλ

k,n Sλ n

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Method 1: Restriction-Induction

Tensor Identity: tensoring with the permutation module is the same as restriction and induction Sλ

n ⌦ Mn ⇠

= IndSn

Sn−1ResSn Sn−1(Sλ n)

⇠ = IndSn

Sn−1

M

ν=λ⇤

Sν

n1

⇠ = M

µ=ν+⇤

M

ν=λ⇤

Sµ

n

Sλ

n ⌦ Mn ⇠

= M

µ=(λ⇤)+⇤

Sµ

n

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Bratteli Diagram: B(S5, S4) for M⌦k

5 M⊗0

5

1 M⊗.5

5

1 M⊗1

5

1 1

M⊗1.5

5

2 1

M⊗2

5

2 3 1 1

M⊗2.5

5

5 5 1 1

M⊗3

5

5 10 6 6 2 1

M⊗3.5

5

15 22 8 9 1

. . . . . . . . . . . . . . . . . .

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Vacillating Tableaux

A vacillating tableaux of shape ` n is a sequence of partitions for which i+ 1

2 = i ⇤

and i+1 = i+ 1

2 + ⇤.

Example: A vacillating tableaux of length 6 and shape :

1 2 3 4 5 6

The multiplicity of Sλ

n in M⌦k n

is given by mλ

k,n = # vacillating tableaux of length k and shape .

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Method 2: Decompose M⌦k

n

into Permutation Modules

I Diagonal Action: ( va ⌦ va ⌦ vb ⌦ va ⌦ vb ⌦ vc ⌦ vd ⌦ vc ) = vσ(a) ⌦ vσ(a) ⌦vσ(b) ⌦ vσ(a) ⌦vσ(b) ⌦ vσ(c) ⌦vσ(d) ⌦ vσ(c)

1 2 3 4 5 6 7 8

I Partition tensor positions: P = {1, 2, 4 | 3, 5 | 6, 8 | 7}: vij = vi` iff j ⇠ ` in P I As a, b, c, d vary over distinct elements of {1, . . . , n}, these simple tensors span a submodule isomorphic to the permutation module M(n4,1,1,1,1) = IndSn

Sn−4⇥S1⇥S1⇥S1⇥S1(1).

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Decompose M⌦k

n

into Permutation Modules

M⌦k

n

⇠ =

n

M

t=0

⇢k t

M(nt,1t) ⇠

=

n

M

t=0

⇢k t M

λ`n

Kλ,(nt,1t)Sλ

n

⇠ =

n

M

t=0

M

λ`n

⇢k t

fλ/(nt)Sλ

n

I ⇢k t

= # set partitions of {1, . . . , k} into t subsets (Stirling 2nd )

I Kλ,(nt,1t) = Kostka number = #semistandard tableaux of shape filled with 0, . . . , 0 | {z }

nt

, 1, 2, . . . , t. I Example:

0 0 0 1 2 3 4 5 6 7 8 9

has = (5, 4, 2, 1), n = 12, t = 9.

I Kλ,(nt,1t) = fλ/(nt) = # standard tableaux shape /(n t)

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Decompose M⌦k

n

into Permutation Modules

M⌦k

n

⇠ = M

λ`n n

M

t=0

⇢k t

fλ/(nt)Sλ

n

mλ

k,n = n

X

t=0

⇢k t

fλ/(nt) = #

⇢ (P, T)

P = partition of {1, . . . , k} into t parts

T= standard tableau of shape /(n t)

Example. A standard set-partition tableau of shape = (5, 4, 2, 1)

P = {1, 6|4, 7, 9, 10|2, 11, 12|8, 14|15, 16|5, 13, 18|3, 17, 19|20} t = 8. n = 12 1, 6 4, 7, 9, 10 2, 11, 12 8, 14 15, 16 5, 13, 18 3, 17, 19 20 T =

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Set-Partition Tableaux ! Vacillating Tableaux

mλ

k,n = #

8 < : Standard set-partition tableaux of shape /(n k) 9 = ; = # 8 < : Vacillating tableaux

f length k

and shape 9 = ; 8 < : 6 4 27 135 9 = ; ! (

1 2 3 4 5 6 7 )

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6 4 27 135

!

1 2 3 4 5 6 7

[BH’19] H-Benkart, [COSSZ’19] Colmenarejo, Orellana, Saliola, Schilling, Zabrocki

6 4 27 135 6 2 4 135 2 4 135 13 2 4 2 13 6 4 135 → 27 2 4 135 → 6 2 4 → 135 13 2 → 4 1 2 1 2 →13 1 → 2 → 1

1. Remove box containing

m = max(T)

2. Delete m from the box
3. Schensted insert box

in T>1

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II. Schur-Weyl Duality and the Partition Algebra

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Centralizer Algebra of Sn on M⌦k

n

Centralizer Algebra: Zk,n := EndSn(M⌦k

n ) =

n 2 End(M⌦k

n )

(x) = (x), 2 Sn
Schur-Weyl Duality:

M⌦k

n

⇠ = M

λ`n

mλ

k,nSλ n

| {z }

as an Sn-module

⇠ = M

λ`n

fλZλ

k,n

| {z }

as a Zk,n-module

I mλ

k,n = multk(Sλ n) = dim(Zλ k,n) =

#(Standard Set-Partition Tableaux) I fλ = dim(Sλ

n) = mult(Zλ k,n) = # (Standard Tableaux)

Artin-Wedderburn theory: dim(Zk,n) = X

λ`n

(mλ

k,n)2

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Bratteli Diagram: B(S6, S5) = B(Zk,6)

k = 0

1 k = .5

1 k = 1

1 1

k = 1.5

2 1

k = 2

2 3 1 1

k = 2.5

5 5 1 1

k = 3

5 10 6 6 2 1 1

k = 3.5

15 22 9 9 2 1

Sum of Squares (Bell No’s) 1 1 2 5 22 + 32 + 11 + 12 =15 52 203 876

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Partition Algebra Pk(n)

[P.P. Martin, V.F.R. Jones, ⇡1993]

Basis of set partitions of {1, . . . , k, 10, . . . , k0}. 1 10 2 20 3 30 4 40 5 50 6 60 7 70 8 80 = ⇢ {1, 3, 40}, {2, 10}, {4, 5, 7}, {6, 80}, {8, 60}, {20, 30}, {50, 70}

Multiplication given by diagram concatenation:

= n Generated by 3 types of diagrams: , ,

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Action of Pk(n) on Tensor Space M⌦k

n

Transposition: u1 u2 u3 u6 v7 v8 u5 u4 u1 u2 u3 u4 u5 u6 u7 u8 u1 u2 u3 u4 u6 v7 v8 v u1 u2 u3 u4 u5 u6 u7 u8 v =

n

X

i=1

vi

projection onto trivial module

δu4,u5 u1 u2 u3 u6 v7 v8 u4 u4 u1 u2 u3 u4 u5 u6 u7 u8 I Commutes with Sn: Pk(n) ! EndSn(M⌦k

n )

I is surjective I it is injective if n 2k the stable case I kernel? [Benkart-H’19]

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Irreducible Modules for the Partition Algebra

Schur-Weyl Duality: M⌦k

n

⇠ = M

λ`n

mλ

k,nSλ n

| {z }

as an Sn-module

⇠ = M

λ`n

fλPλ

k

| {z }

as a Pk(n)-module

I mλ

k,n = dim(Pλ k) = #(Standard Set-Partition Tableaux)

I fλ = dim(Sλ

n) = # (Standard Tableaux)

The irreducible partition algebra module: Pλ

k = C-span

⇢ vT

T 2

⇢standard set-partition tableaux of shape Question Is there a combinatorial action analogous to Young’s repre- sentations of Sn on standard tableaux?

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Action on Basis Indexed by Set-Partition Tableaux

[H-Jacobson, 2018] · · · 3, 5, 6 11 1, 2 4 12 8, 9, 10 7, 13 1 2 3 4 5 6 7 8 9 10 11 12 13 4 10 13 = · · · 1, 2, 3 8, 11 9 5, 6, 7 12 4 10, 13 · · · 3, 5, 6 11 1, 2 4 12 8, 9, 10 7, 13 1 2 3 4 5 6 7 8 9 10 11 12 13

= · · · 5 8, 9 10, 12, 13 1, 2, 4 3, 6 1, 2, 4 7, 11 = 0

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Action of Generators

, ,

· · · 3, 5 1, 2 4 6 8 7, 9 1 2 3 4 5 6 7 8 9 = · · · 1, 3 2, 4 5 7 6 8, 9

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Action of Generators

, ,

· · · 3, 5 1, 2 4 6 8 7, 9 1 2 3 4 5 6 7 8 9

= · · · 2 3, 5 1 4 6 8 7, 9

· · · 3, 5 1, 2 4 6 8 7, 9 1 2 3 4 5 6 7 8 9

= · · · 4 3, 5 1, 2 6 8 7, 9 = 0

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Action of Generators

, ,

· · · 3, 5 1, 2 4 6 8 7, 9 1 2 3 4 5 6 7 8 9

= · · · 1,2,3,5 4 6 8 7, 9

· · · 3, 5 1, 2 4 6 8 7, 9 1 2 3 4 5 6 7 8 9 = · · · 3, 5 1, 2 4 8 6,7,9 = 0

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Restricts Naturally to Subalgebras of the Partition Algebra

Brauer Algebra (matchings)

· · · 1, 5 6, 9 2 4 7 3 8

Rook Brauer Algebra (partial matchings)

· · · 2 1, 4 5 6, 8 3 9 7

Temperley-Lieb (planar matchings)

· · · 2, 3 1, 5 8, 9 4 6 7

Motzkin Algebra (planar partial matchings)

· · · 3, 4 5 8 7, 9 1 2 6

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Restricts Naturally to Subalgebras of the Partition Algebra

Symmetric Group (permutations)

· · · 1 2 5 8 3 4 9 6 7

Rook Monoid (partial permutations)

· · · 3 4 5 7 1 2 8 6 9

Identity (planar permutations)

1 2 3 4 5 6 7 8 9 · · ·

Planar Rook Monoid (planar partial permutations)

· · · 1 2 3 5 6 9 4 7 8

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III. Insertion Bijections

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III. Insertion Bijections:

dim(P

k(n)) =

X

`n

(m

k,n)2

1 2 3 4 5 6 1 2 3 4 5 6

@

1,3 2 4 5,6

,

5 2,3 6 1,4

1 A P = ⇣

1 2 3 4 5 6

⌘ Q = ⇣

1 2 3 4 5 6

⌘

[COSSZ’19] [HL’04] [CDDSY’06] [BH’19]∗ [COSSZ’19]

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[HL’04] [COSSZ’19]

1 2 3 4 5 6 1 2 3 4 5 6

5 2,3,7,8 6,9 1,4,11 10,12 5 2,3,7,8 6,9 1,4,11 10 5 2,3,7,8 6,9 1,4 10 5 2,3,7,8 6,9 1,4 5 2,3,7,8 1,4 6 5 1,4 6 2,3,7 5 2,3 1,4 6 5 2,3,7,8 6,9 1,4,11 10,12 5 2,3,7,8 6,9 10 1,4,11 5 2,3,7,8 6,9 1,4 10 5 2,3,7,8 1,4 6,9 5 1,4 6 2,3,7,8 5 1,4 6 2,3,7

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[COSSZ’19]

✓

1,3 2 4 5,6

,

5 2,3 6 1,4

◆

1 2 3 4 5 6 1 2 3 4 5 6

Generalized permutation of propagating blocks:

!

✓{2, 3} {1, 4} {6} {5, 6} {2} {4} ◆ non-propagating blocks: {5}, {1, 3} Insertion Tableau Recording Tableau ; ; {5, 6}!

5,6 2,3

{2} !

2 5,6 2,3 1,4

{4} !

2 4 5,6 1,3 2,3 6 1,4 5

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Many Nice Properties

here are a few

1. Reflect the diagram over its horizontal axis:

d =

!

(P, Q) flip(d) =

!

(Q, P)

2. Consequence: |{symmetric diagrams}| = P

λ dim(Pλ k).

Implies: a “model” representation on symmetric diagrams ([H-Reeks’15]).

3. Respects subalgebras:

$

· · · 1 2 5 8 3 4 9 6 7

$

· · · 1, 5 6, 9 2 4 7 3 8

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A Few References

[BH] G. Benkart and TH, Partition Algebras and the Invariant Theory of the Symmetric Group, Recent Trends in Algebraic Combinatorics, Springer/AWM (2019). [CDDSY] W. Chen, E. Deng, R. Du, R. Stanley and C. Yan, Crossings and nestings of matchings and partitions, Transactions of the AMS (2006). [COSSZ] L. Colmenarejo, R. Orellana, F. Saliola, A. Schilling, and M. Zabrocki, An insertion algorithm on multiset partitions with applications to diagram algebras, arXiv:1905.02071 (2019). [HL] TH and T. Lewandowski, RSK insertion for set partitions and diagram algebras, Electronic J. Combinatorics (2004/6). [HL] TH and M. Reeks, Gelfand Models for Diagram Algebras, JACo (2015). [HJ] TH and T.N. Jacobson, Set-partition tableaux and representations of diagram algebras arXiv:1808.08118 (2018).